720 research outputs found
A Posteriori Error Control for the Binary Mumford-Shah Model
The binary Mumford-Shah model is a widespread tool for image segmentation and
can be considered as a basic model in shape optimization with a broad range of
applications in computer vision, ranging from basic segmentation and labeling
to object reconstruction. This paper presents robust a posteriori error
estimates for a natural error quantity, namely the area of the non properly
segmented region. To this end, a suitable strictly convex and non-constrained
relaxation of the originally non-convex functional is investigated and Repin's
functional approach for a posteriori error estimation is used to control the
numerical error for the relaxed problem in the -norm. In combination with
a suitable cut out argument, a fully practical estimate for the area mismatch
is derived. This estimate is incorporated in an adaptive meshing strategy. Two
different adaptive primal-dual finite element schemes, and the most frequently
used finite difference discretization are investigated and compared. Numerical
experiments show qualitative and quantitative properties of the estimates and
demonstrate their usefulness in practical applications.Comment: 18 pages, 7 figures, 1 tabl
Colour image segmentation by the vector-valued Allen-Cahn phase-field model: a multigrid solution
We propose a new method for the numerical solution of a PDE-driven model for
colour image segmentation and give numerical examples of the results. The
method combines the vector-valued Allen-Cahn phase field equation with initial
data fitting terms. This method is known to be closely related to the
Mumford-Shah problem and the level set segmentation by Chan and Vese. Our
numerical solution is performed using a multigrid splitting of a finite element
space, thereby producing an efficient and robust method for the segmentation of
large images.Comment: 17 pages, 9 figure
A discrete approximation of Blake & Zisserman energy in image denoising and optimal choice of regularization parameters
We consider a multi-scale approach for the discrete approximation of a functional proposed by Bake and Zisserman (BZ) for solving image denoising and segmentation problems. The proposed method is based on simple and effective higher order varia-tional model. It consists of building linear discrete energies family which Γ-converges to the non-linear BZ functional. The key point of the approach is the construction of the diffusion operators in the discrete energies within a finite element adaptive procedure which approximate in the Γ-convergence sense the initial energy including the singular parts. The resulting model preserves the singularities of the image and of its gradient while keeping a simple structure of the underlying PDEs, hence efficient numerical method for solving the problem under consideration. A new point to make this approach work is to deal with constrained optimization problems that we circumvent through a Lagrangian formulation. We present some numerical experiments to show that the proposed approach allows us to detect first and second-order singularities. We also consider and implement to enhance the algorithms and convergence properties, an augmented Lagrangian method using the alternating direction method of Multipliers (ADMM)
A Relaxation result for energies defined on pairs set-function and applications
We consider, in an open subset Ω
of RN, energies depending on the perimeter of a subset
E С Ω
(or some equivalent surface integral) and on a function u which is defined only on
E. We compute the lower semicontinuous envelope of such energies. This relaxation has
to take into account the fact that in the limit, the “holes”
Ω \ E may collapse into a
discontinuity of u, whose surface will be counted twice in the relaxed energy. We discuss
some situations where such energies appear, and give, as an application, a new proof of
convergence for an extension of Ambrosio-Tortorelli’s approximation to the Mumford-Shah
functional
A Variational Shape Optimization Approach for Image Segmentation with a Mumford-Shah Functional
We introduce a novel computational method for a Mumford–Shah functional, which decomposes a given image into smooth regions separated by closed curves. Casting this as a shape optimization problem, we develop a gradient descent approach at the continuous level that yields nonlinear PDE flows. We propose time discretizations that linearize the problem and space discretization by continuous piecewise linear finite elements. The method incorporates topological changes, such as splitting and merging for detection of multiple objects, space–time adaptivity, and a coarse-to-fine approach to process large images efficiently. We present several simulations that illustrate the performance of the method and investigate the model sensitivity to various parameters
Joint Image Reconstruction and Segmentation Using the Potts Model
We propose a new algorithmic approach to the non-smooth and non-convex Potts
problem (also called piecewise-constant Mumford-Shah problem) for inverse
imaging problems. We derive a suitable splitting into specific subproblems that
can all be solved efficiently. Our method does not require a priori knowledge
on the gray levels nor on the number of segments of the reconstruction.
Further, it avoids anisotropic artifacts such as geometric staircasing. We
demonstrate the suitability of our method for joint image reconstruction and
segmentation. We focus on Radon data, where we in particular consider limited
data situations. For instance, our method is able to recover all segments of
the Shepp-Logan phantom from angular views only. We illustrate the
practical applicability on a real PET dataset. As further applications, we
consider spherical Radon data as well as blurred data
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